Thermodynamic Probabilities

Inspection of equation (6.2) reveals that for the entropy of each state, there is apparently a relation 

This is the equation made famous by Boltzmann in 1896, and inscribed on his tombstone in Vienna. It of course has a more rigorous derivation than the one we have given. 

Now for this particular example of 4 molecules and 4 energy levels there are 4 = 256 possible microstates and 35 configurations. Some of these configurations might be 

In the present case we can use our knowledge about the relation between entropy and randomness and predict a priori that 4 configurations. 

SO it should have the highest entropy and occur most often. Applying equation (6.4), the thermodynamic probabilities, W, of these two limiting configurations are 

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A quantitative way of dealing with the degree of disorder in a system is to define something called the thermodynamic probability Q. which counts the number of ways in which a particular state can come about. Thus situations we characterize as relatively disordered can come about in more ways than a relatively ordered state, just as an unordered deck of cards compared to a deck arranged by suits. 

Chemists learn to use the thermodynamic probability almost instinctively in a qualitative manner it is quantitatively related to entropy through an equation due to Boltzmann ... 

For the evaporation process we mentioned above, the thermodynamic probability of the gas phase is given by the number of places a molecule can occupy in the vapor. This, in turn, is proportional to the volume of the gas (subscript g) 12- oc V In the last chapter we discussed the free volume in a liquid. The total free volume in a liquid is a measure of places for molecules to occupy in the liquid. The thermodynamic probability of a liquid (subscript 1) is thus V, oc V, frgg. Based on these ideas, the entropy of the evaporation process can be written as... 

A moment s reflection will convince us that these probabilities can be used as thermodynamic probabilities in Eq. (3.21) to calculate the entropy change on stretching ... 

The thermodynamic probability is converted to an entropy through the Boltzmann equation [Eq. (3.20)] so we can write for the entropy of the mixture (subscript mix)... 

If the polymer system was able to exist in an equilibrium state only, then a strictly defined correlation between (a, ph) and (a, ph) would exist in particular conditions, according to minimum of free energy of system formation. Consequently, there would occur only one temperature at which process initiation is thermodynamically probable. In rare ca.ses there may occur different correlations between ( ph, a) and ( ph, a ), which display one and the same value of free energy minimum of system formation. 

Concluding from the last expression, the temperature at which reaction initiation is thermodynamically probable, is determined from the equation ... 

In equation (1.17), S is entropy, k is a constant known as the Boltzmann constant, and W is the thermodynamic probability. In Chapter 10 we will see how to calculate W. For now, it is sufficient to know that it is equal to the number of arrangements or microstates that a molecule can be in for a particular macrostate. Macrostates with many microstates are those of high probability. Hence, the name thermodynamic probability for W. But macrostates with many microstates are states of high disorder. Thus, on a molecular basis, W, and hence 5, is a measure of the disorder in the system. We will wait for the second law of thermodynamics to make quantitative calculations of AS, the change in S, at which time we will verify the relationship between entropy and disorder. For example, we will show that... 

For polycarbonates this was checked to be a good approximation [43,195], Since each distribution function P is determined at each temperature separately, the individual p values for different temperatures give the thermodynamic probabilities of a given state, which simply means that ... 

Function 0)(s) determines the probability that the RW trajectory of the random walk is simultaneously also by SARJV trajectory and by its own last step realizes the state M(s). Hence, it is numerically equal to part of these SARW trajectories among general number (Idf of RW trajectories which realize the state M(s). Number L(s) of such SARJV trajectories determines the thermodynamical probability of the realization M(s) ... 

In continuous space the thermodynamical probability W(x) of the realization of state in... 

The derivation considers the number of ways to, of placing each of the n segments in, say, the ith solute molecule, then goes on to count the number of ways of scaling this up for TV2 solute molecules. The thermodynamic probability of the mixture Qmix results from these steps. 

When applied to a system with a large number of elements, the distributions are measured by thermodynamic probability (W) the most probable distribution is such that IV is a maximum. This optimal principle is consistent with tile condition of maximum entropy (5) cited... 

The transitory poisoning by scavengers is explained by competitive adsorption of a halogen-containing species on catalyst sites that are needed for the oxidation of CO and hydrocarbons. In the case of EDB it is thermodynamically probable that HBr 33), or Br2 is the actual adsorbed species (66). The possible interactions of EDB and EDC with TEL and the resulting loss in noble metal surface area on the one hand, and catalyst activity on the other, are very complex (66). 

Notice that the direction of the process and time have been determined This has been called the arrow of time [2], Time proceeds in the direction of entropy generation, that is, toward a state of greater probability for the total of the system and its environment, which, in the widest sense, makes up the universe. Finally, we wish to point out that an interesting implication of Equation 2.10 is that for substances in the perfect crystalline state at T = 0 K, the thermodynamic probability Q = 1 and thus S = 0. 

Finally, equilibrium processes can be defined as processes between and passing states that all have the same thermodynamic probability. On the one hand, these processes proceed without driving forces on the other hand, and this is inconsistent and unrealistic, there is no incentive for the process to proceed. These imaginary processes function only to establish the minimum amount of work required, or the maximum amount of work available, in proceeding from one state to the other. 

Indeed, for different molecules, which otherwise are nearly the same, such as isomers, or molecules of about the same size, polarity, or other properties, the thermodynamic probability of the mixed state at the same P and T is much larger than that of the respective pure states (in molar units) ... 

Using the taken symbols, we get the formulae for configuration energy Ei and thermodynamic probability Wi ... 

Clearly the above scheme of liquid-phase oxidation by oxygen shows H202 and ROOH formed as intermediate products. However, they cannot be related to active sites of another, secondary reaction as is customary in, for example, conjugated processes. The reasons for making such comparisons are as follows firstly, H202 and ROOH are final products of a complex reaction and initial reagents for other thermodynamically probable reactions secondly, their formation and consumption (by the scheme selected) do not correspond to the notion of active site of conjugated reactions. 

Planck s hypothesis is based on the statistical thermodynamics in that the entropy is related to the number of possible energy states for a given energy or thermodynamic probability W ... 

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